Quadratic Form Index

The index I associated to a symmetric, non-degenerate, and bilinear g over a finite-dimensional vector space V is a nonnegative integer defined by

 I=max_(W in S)(dimW)

where the set S is defined to be

 S={W:W is a subspace of V and g|_W is negative definite}.

As a concrete example, a pair (M,g) consisting of a smooth manifold M with a symmetric (0,2) tensor field g is said to be a Lorentzian manifold if and only if dimM>=2 and the index I associated to the quadratic form gx satisfies I=1 for all x in M (Sachs and Wu 1977). This particular definition succinctly conveys the fact that Lorentzian manifolds have indefinite metric tensors of signature (1,n-1) (or equivalently (n-1,1)) without having to make precise any definitions related to metric signatures, quadratic form signatures, etc.

The above example also illustrates the deep connection between the index of a quadratic form and the notion of the index of a metric tensor g defined on a smooth manifold M. In particular, the index of a metric tensor g is defined to be the quadratic form index associated to gx for any element x in M. Because of this connection, indices are especially significant to various fields: In some literature on differential and Riemannian geometry, for example, the notion of an index is used as a main tool by which to define a metric tensor (Sachs and Wu 1977).

See also

Lorentzian Manifold, Metric Signature, Metric Tensor, Metric Tensor Index, Quadratic Form Signature, Smooth Manifold

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Sachs, R. K. and Wu, H. General Relativity for Mathematicians. New York: Springer-Verlag, 1977.

Cite this as:

Weisstein, Eric W. "Quadratic Form Index." From MathWorld--A Wolfram Web Resource.

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